Demand reduction and initial endowments in consignment auctions

Technical Summary: Demand Reduction and Initial Endowments in Consignment Auctions

Problem Statement
The paper investigates the strategic behavior and equilibrium outcomes in consignment auctions, a mechanism frequently employed in emissions allowance trading (cap-and-trade) markets. In this two-stage process, bidders (typically pollution-emitting firms) first receive free initial endowments of a good (e.g., emissions permits) from a regulator. In the second stage, these bidders must consign their initial endowments back to the auctioneer, who then sells the pooled supply (initial endowments plus any additional units provided by the auctioneer) via a uniform-price auction to the same set of bidders.

The core theoretical challenge addressed is the interaction between quantity constraints and initial endowments. Unlike standard auctions where bidders are purely buyers, participants in consignment auctions act as both buyers (vying for additional units) and sellers (receiving revenue for their consigned endowments). The paper specifically models bidders with flat demands up to a quantity constraint and constant marginal values, examining how the presence of initial endowments affects demand reduction, equilibrium prices, and welfare.

Methodology
The author employs a game-theoretic approach to characterize the equilibrium of uniform-price consignment auctions. The analysis proceeds in two main phases:

1. Equilibrium Characterization:

   • The paper first assumes common knowledge of valuations ($v_i$) among bidders to derive a Nash equilibrium.
   • It then demonstrates that this outcome is equivalent to the dominant strategy equilibrium of a specific ascending auction format (a variant of the English auction) where bidders have private information about their valuations.
   • An iterative procedure is developed to determine the equilibrium price and allocation. This procedure accounts for the "endowment effect," where a bidder's maximum bid is influenced not only by their valuation but also by the revenue they would lose if they fail to secure their consigned units at a high price. The procedure involves calculating a maximum bid threshold ($\hat{b}$) for each bidder, which depends on their endowment ($e_i$), quantity constraint ($q_i$), and the valuations of other bidders.

2. Comparative Static Analysis:

   • Using the characterized equilibrium, the paper analyzes how changes in parameters—specifically initial endowments ($e_i$) and total supply ($m$)—affect the equilibrium price, fulfilled demand, social welfare, and auctioneer revenue.
   • The analysis distinguishes between "proportional consignment" (where total endowments are less than total supply) and "pure consignment" (where total endowments equal total supply).

Key Contributions

• Theoretical Framework for Consignment Auctions: The paper extends existing auction theory (specifically Yoon, 2026) to the specific context of consignment auctions with quantity constraints. It highlights that the presence of initial endowments fundamentally alters the bidding incentives, creating a trade-off between shading bids as a buyer and raising offers as a seller.
• Mechanism of Demand Reduction: The author characterizes conditions under which demand reduction (bidders bidding lower to secure a residual quantity at a lower price) leads to a low-price equilibrium. The paper argues that such low-price equilibria are not exceptions but can be the rule, particularly when no single bidder's quantity constraint is large enough to cover the total supply.
• Analytical Characterization of Endowment Effects: The paper provides a rigorous derivation of how increasing initial endowments affects the maximum bid function, showing that endowments can act as a catalyst to prevent low-price equilibria.

Key Results

1. Effect of Increasing Initial Endowments ($e_i$):

   • Price: An increase in a bidder's initial endowment leads to a higher equilibrium price. This occurs because the endowment increases the bidder's incentive to bid aggressively; they stand to lose more revenue from their consigned units if the price is low.
   • Demand: The bidder's fulfilled demand increases.
   • Revenue and Welfare: The effects on the auctioneer's revenue and social welfare are ambiguous.
     • Revenue: While intuition suggests revenue should fall because the auctioneer sells fewer units, the paper shows revenue can increase. This happens if the increase in endowments prevents a low-price equilibrium, causing the price to jump significantly (e.g., from zero to a positive valuation). The "price increase effect" can outweigh the "quantity decrease effect."
     • Welfare: Social welfare may increase or decrease. It tends to increase if the prevention of a low-price equilibrium leads to a final allocation that better aligns with bidders' marginal values.

2. Effect of Increasing Total Supply ($m$):

   • Price: An increase in total supply leads to a lower equilibrium price, consistent with standard supply-demand logic.
   • Revenue and Welfare: Similar to the endowment case, the effects on revenue and social welfare are ambiguous and depend on the specific configuration of bidders' constraints and valuations.

3. Concurrent Increases: If both the initial endowment and total supply increase by the same amount, the equilibrium price decreases (driven by the supply effect), and the auctioneer's revenue decreases, while social welfare remains ambiguous.

Significance and Claims
The paper claims its primary contribution is the analytical examination of consignment auctions under quantity constraints, offering comparative static results useful for the design and regulation of cap-and-trade markets.

• Policy Implication: The findings suggest that increasing the proportion of permits allocated freely (initial endowments) does not necessarily reduce auctioneer revenue. On the contrary, it can enhance revenue by mitigating the strategic demand reduction that drives prices down.
• Alignment with Empirics: The theoretical result that higher initial endowments lead to higher fulfilled demands aligns with empirical observations from the RECLAIM market (Fowlie and Perloff, 2013).
• Distinction from Standard Auctions: The paper emphasizes that the "endowment effect" creates a unique dynamic where bidders are not purely buyers, distinguishing consignment auctions from standard uniform-price auctions and partnership resolution games.

The author remains modest regarding the scope, noting that while the model assumes flat demands and constant marginal values (a common assumption in electricity and emissions markets), the results provide a foundational understanding of how endowment structures influence market outcomes in environmental regulation.